Counting Frabjous Numbers
A 'frabjous' number is a 3-digit number where all digits are odd, and no adjacent digits are the same.
The odd digits are: 1, 3, 5, 7, 9. There are 5 odd digits available.
We need to determine the number of choices for each of the three positions (hundreds, tens, units).
- Hundreds digit: This digit must be odd. There are 5 choices (1, 3, 5, 7, 9).
- Tens digit: This digit must also be odd, but it cannot be the same as the hundreds digit. Since one odd digit is used for the hundreds place, there are $5 - 1 = 4$ choices remaining.
- Units digit: This digit must be odd and cannot be the same as the tens digit. It can be the same as the hundreds digit. Since one odd digit is used for the tens place, there are $5 - 1 = 4$ choices remaining.
Calculating Total Frabjous Numbers
To find the total number of possible frabjous numbers, we multiply the number of choices for each digit:
Total Numbers = (Choices for hundreds digit) $\times$ (Choices for tens digit) $\times$ (Choices for units digit)
Total Numbers = $5 \times 4 \times 4$
Total Numbers = $80$
Therefore, there are 80 such frabjous numbers.
Important Questions from Numerical Reasoning
- Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{th}$ stair. For example, $F(1) = 1$, $F(2) = 2$, $F(3) = 3$. The value of $F(5)$ is ________.- In a zoo, three lions and four tigers eat 390 kg of food every week. In another zoo, four lions and five tigers eat 500 kg of food every week. Lions and tigers eat different amounts of food, but all individuals of the same species eat the same amount. The amount of food a single lion eats per week is ________ kg.
(Answer in integer) Consider a spherical globe rotating about an axis passing through its poles. There are three points P, Q, and R situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let P, Q, and R move with speeds $v_P$, $v_Q$, and $v_R$, respectively.
Which one of the following options is CORRECT?
A dozer pushes up a 100 kg spool of cable along a $20^ \circ$ incline road at a constant velocity as shown in the figure. The figure shows a dozer pushing a spool (diameter 400 mm) up a $20^ \circ$ incline. Point A is the contact point between the spool and the road, and Point B is the contact point between the dozer bucket and the spool. The coefficient of static friction between the dozer bucket and the spool (Point B) is 0.45, and coefficient of kinetic friction between road and the spool (Point A) is 0.15.
Consider the spool only slides up the incline. The maximum normal force in N acting at Point B, is ___________ [rounded off to 1 decimal place]
